Models with three Higgs doublets in the triplet representations of A4 or S4

TL;DR

This paper demonstrates that models with three Higgs doublets in triplet representations of A4 or S4 cannot simultaneously produce realistic quark masses and CP violation, limiting their viability.

Contribution

It proves that such models cannot generate both non-zero quark masses and CP violation in the CKM matrix for all possible field representations and vacuum alignments.

Findings

Cannot obtain non-zero quark masses and CP violation simultaneously

All vacuum alignments lead to vanishing quark masses or CP violation

Models with three Higgs triplets in A4 or S4 are incompatible with experimental quark data

Abstract

We consider the quark sector of theories containing three scalar SU(2)_L doublets in the triplet representation of A4 (or of S4) and three generations of quarks in arbitrary A4 (or S4) representations. We show that, for all possible choices of quark field representations and for all possible alignments of the Higgs vacuum expectation values that can constitute global minima of the scalar potential, it is not possible to obtain simultaneously non-vanishing quark masses and a non-vanishing CP-violating phase in the CKM quark mixing matrix. As a result, in this minimal form, models with three scalar fields in the triplet representation of A4 or of S4 cannot be extended to the quark sector in a way consistent with experiment.